Many products use film coatings to modify the characteristics of the product's surface. Polycarbonate eyeglass lenses, for example, use a film “hard coat” to protect against scratching Film thicknesses can range from 0.0001 micron (um) to hundreds of microns, depending upon the application. It is usually important to control the film's thickness and its composition, whether to optimize the performance of the film or simply to minimize the amount of film material that is used.
Spectral reflectance is common method for measuring film thickness, as well as for measuring other film properties such as optical constants. Spectral reflectance methods first measure the amount of light reflected off of or transmitted through the film sample (which contains the film or films of interest, along with any other films or substrate that might be present) over a range of wavelengths, and then analyze this reflectance spectrum to determine the film's properties.
The term “spectral reflectance” (SR) is used herein to refer to both reflectance and transmittance measurements, at both normal and non-normal incident angles, where polarization changes neither occur nor are detected. This distinguishes SR from ellipsometry, which, in contrast, is concerned primarily with polarization changes induced by the sample. For examples and general reference, see “Spectroscopic Ellipsometry and Reflectometry: A User's Guide” by Tompkins and McGahan, John Wiley & Sons, 1999. Companies such as Filmetrics, Inc. of San Diego, Calif. manufacture such spectral reflectance systems.
The analysis of a measured SR spectrum typically consists of comparing it to a set of theoretical SR spectra that are generated based on the sample's expected properties. Some of these properties are known and thus fixed when generating the theoretical spectra, while others are unknown (i.e. they are being measured) and assume a range values corresponding to those anticipated in the sample. The values of the unknown properties that result in the theoretical spectrum that most closely matches the measured spectrum are then taken to most closely represent the actual properties of the sample; these values are thus the “measured” values.
There are a number of film properties that might need to be measured. These include the film's thickness, roughness, and composition. The composition, which may be homogenous or not, is usually described in terms of the two optical constants: refractive index (n) and extinction coefficient (k). Both n and k are functions of wavelength, which means that, when measured, they must be determined for each discrete wavelength that comprises the SR spectrum. If the number of discrete wavelength data points in the SR spectrum is represented as i, then the number of values of n that need to be determined is i, as is the number of values of k that need to be determined. Even assuming a smooth homogeneous film (i.e. no roughness or compositional grading) of thickness d, the number of free (solved for) parameters is 2i+1 (n and k and thickness), whereas the number of known parameters (R) is only i. Since more known parameters than unknown parameters are required to be able to solve uniquely for a film's properties, either d and k much be known (the latter often=0) to solve for n, or, as is more often the case, n and k are known (or at least assumed) and d is solved for.
The usual method of simultaneously solving for a film's d, n, and k with only i reflectance data points is to use a mathematical model to estimate the wavelength dependence of the film's n and k. Such a model (e.g., Lorentz, Harmonic Oscillator) can reduce the number of parameters required to describe n and k down from 2i to between one and a few dozen. This method works acceptably for films whose n and k are well-described by such a model. However, for many films, this method can result in a loss in accuracy (especially for models with fewer parameters) and solving robustness (especially for models with larger numbers of parameters). Additionally, resorting to mathematical models for n and k is a challenge in the course of day-to-day measurements—the complexity of the models means that the skill of highly-trained experts are often required whenever a new film stack is encountered. This is not usually the case in a “model-free” situation.
A method in the art for acquiring an additional i known parameters (i.e. one additional known parameter per wavelength data point) is to combine transmittance (T) with reflectance measurements, for a total of 2i known parameters. For a smooth (non-scattering) film, this is still fewer than the required 2i+1 unknown parameters (n and k and thickness), so either d must be known or a model used for n and k. Note that when k=0 there is no absorption and T=1−R, so that there are only really i known and i+1 unknown parameters, so that T adds no additional information in this case.